Studentized residual

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing. It is a form of a Student's t-statistic, with the estimate of error varying between points. In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing. It is a form of a Student's t-statistic, with the estimate of error varying between points. This is an important technique in the detection of outliers. It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym Student. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing. The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions. Consider the simple linear regression model Given a random sample (Xi, Yi), i = 1, ..., n, each pair (Xi, Yi) satisfies where the errors ε i {displaystyle varepsilon _{i}} , are independent and all have the same variance σ 2 {displaystyle sigma ^{2}} . The residuals are not the true errors, but estimates, based on the observable data. When the method of least squares is used to estimate α 0 {displaystyle alpha _{0}} and α 1 {displaystyle alpha _{1}} , then the residuals ε ^ {displaystyle {widehat {varepsilon ,}}} , unlike the errors ε {displaystyle varepsilon } , cannot be independent since they satisfy the two constraints